Teaching and Assessing Polygons Using
Technology
T. Soucie MSEd,
Matka Laginje Elementary School
Croatia
R. Svedrec
Otok Elementary School
Croatia
N. Radovic M.S.,
Department of Geodesy
University of Zagreb
Croatia
I. Kokic
Trnsko Elementary School
Croatia
[email protected]
[email protected]
[email protected]
[email protected]
Using dynamic geometry software to teach geometry can transform the way we
approach teaching and learning. Meaningful technology based tasks allow students to
discover mathematics on their own and construct their own understanding. Furthermore, they
require active engagement of students and encourage higher level thinking. They challenge
students to think about mathematical ideas in a new light.
In the following article we will show how technology helped us create an engaging
learning environment for a unit on Polygons as well as share strategies that have helped us
implement technology effectively. Through various age appropriate activities students explore
polygons and their properties, make conjectures, test them, reason about geometric ideas as
well as demonstrate understanding and ability to apply their knowledge of polygons.
For every lesson we prepare a „Sketchpad notebook”. This „notebook” contains basic
theory, a few solved examples (step by step), as well as problems for individual
exploration/learning. Class usually begins with a discussion and a presentation of examples
followed by individual work by students. At that time, we introduce new terminology and
concepts students are not able to discover on their own. Then students work independently or
in pairs on the assigned tasks (see figure 1 and figure 2).
Fig. 1 Students use geometry software to explore the rules for the number of diagonals from
one vertex of a polygon and the sum of diagonals in a polygon
Question 1: Draw the quadrilateral ABCD and all its diagonals. How many are there? Name
them.
Question 2: Draw the convex pentagon ABCDE and all its diagonals. Name them. How many
diagonals start (end) at point A? How many diagonals does a pentagon have?
Question 3: Draw the convex hexagon ABCDEF and all its diagonals. Name them. How many
diagonals start (end) at point C? How many diagonals does a hexagon have?
Question 4: Draw the convex heptagon ABCDEFG and all its diagonals. Name them. How
many diagonals start (end) at point D? How many diagonals does a heptagon have?
Question 5: What is the number of diagonals from one vertex of a polygon?
Question 6: What is the total number of diagonals in a polygon?
Fig. 2 Students use geometry software to explore central angles of a polygon
Problem 1: Construct equilateral triangle ABC and its inscribed and circumscribed circles.
(The center of the circumscribed circle (circumcenter) is the point of intersection of the
perpendicular bisectors of the sides. The center of the inscribed circle (incenter) is the point of
intersection of three internal angle bisectors of a triangle.
Problem 2: Measure the lengths of sides AS, BS i CS and the measures of
angles ASB, BSC i CSA. What do you notice?
Problem 3: Construct square ABCD. Inscribe it and circumscribe it. (The centers of both
circles are at the same point. Label it S.)
Problem 4: Measure the lengths of sides AS, BS, CS
measures of angles ASB, BSC, CSD and DSA. What do you notice?
and
DS
and
the
Problem 5: Regular pentagon ABCDE is given. Point S is a center of its inscribed and
circumscribed circle. Measure the lengths of the sides AS, BS, CS, DS i ES and measures of
the angles ASB, BSC, CSD, DSE i ESA. What do you notice?
D
E
C
S
B
A
Problem 6. Complete the table.
n
5
6
9
20
αn
30º
Problem 7. Inscribe a square in the given circle.
Problem 8. In the given circle inscribe a regular hexagon.
18º
24º
36º
This is an excellent time for the teacher to observe students, the way they approach the
problems, the strategies they are using to solve them and the conjectures they make. Listening
to student interactions while working on problems offers valuable insight into student
understanding. The role of the teacher during these explorations is to provide students with
frequent feedback and guidance as well as intervene and clear up any misconceptions, if
necessary (see figure 3).
Fig. 3 A picture of students exploring propeties of polygons
At the end of class, students save their work on the server. If they did not have enough
time to complete their work, they can continue working on it at home or after school. Once
they have completed their assignment, they email it to their teacher. The teacher assesses
student work and returns it with his/he comments and explanations on how to improve it, if
necessary.
The next class, the teacher can show all the solutions and comment on them with
students. In case students have difficulties solving the problems, the teacher can prepare a
step-by-step presentation to show how the problems were supposed to be solved. These
presentations are also convinient when there is no access to a computer lab (see figure 4).
Fig. 4 Step by step presentations showing various methods of inscribing a regular
hexagon in the circle of the given radius
Once students have adequate knowledge and understanding of polygons, they can
begin to work on various cross-curricular tasks. In addition to building new geometrical ideas,
such tasks allow student to see the application of mathematics in real-life and help them
develop an appreciation for the subject. Here is an example of one of the cross-curricular
tasks we use in this unit.
Penrose tiling
British scientist Sir Roger Penrose came up with a special tiling that uses two shapes, a
kite and a dart. The tiles must be placed so that each vertex with a dot always touches only other
vertices with dot. These tiles make a non-periodic tiling, and do not repeat by translations.
Penrose tiling created in Sketchpad
Copy the two tiles shown below – the kite and the dart with their dots – on „Sketchpad paper“.
Create your unique Penrose tiling. Answer the questions below:
1. How many different combinations of kites and darts can surround a vertex point?
2. Explore how the dots affect the tiling. Why the dots are important (see figure 5)?
(Serra, 2008)
Fig. 5 Samples of student work
Technology enables us to individualize instruction because students are able to work at
their own pace. They can revisit some previous activities for quick reminders or explore
certain concepts more, if they need to. While some students need extra time to complete the
assigned tasks, others will be done quicker. Enrichment activities can be offered to those
students who complete their task successfully (see figure 6).
Enrichment activity – Iteration
Example 1
Using iterations, construct a regular polygon with the given length of a side.
Solution:
1. Define parameter n (Use the Number menu to select the command New Parameter.).
2. Construct segment AB .
(n − 2) ⋅ 180° ⎞
3. Calculate the exterior angle of the regular n – gon ⎛⎜ 180° −
⎟
n
⎝
⎠
Use the Number menu to select the command Calculate and write as in the picture.
(n − 2) ⋅ 180°
180° −
n
Note: For n you select (click on) the parameter n that you constructed in step 1.
4. Translate point A for vector BA.
(First select point B and then point A. Use the Transform menu to select the command
Mark vector. Next, select point A and use the Transform menu to select the command
Translate. Name the new point A'.)
5. Rotate point A' around point A by the angle calculated in step 3.
(Double-click on point A, then select the angle calculated in step 3. Use the Transform
menu to select the command Mark angle. Select point A' and use the Transform menu to
select the command Rotate. Name the new point A''.)
6. Hide point A'. (Use the Display menu to select the command Hide Point.)
7. Select points A and B and parameter n. Hold the shift key and use the Transform menu to
select the command Iterate to Depth. In the new window, select the image for points A
and B.
For point A click on point A'' and for point B click on point B. Finish by clicking on
Iterate.
8. Click + or – on parameter n to change the type of regular polygon.
Example 2
Using iterations, construct a regular polygon if the radius of a circumscribed circle is given.
Solution:
1. Define parameter n (Use the Number menu to select the command New Parameter.)
2. Construct a circle and hide the point on the circle.
3. Construct a new point on the circle and label it A (Select the circle and use the Construct
menu to select the command Point on Circle.).
360° ⎞
4. Calculate the central angle of the regular n – gon ⎛⎜
⎟.
⎝ n ⎠
5. Rotate point A around point S by the angle calculated in step 4 and label the new point A'.
6. Draw segment AA ' .
7. Select point A and parameter n. Hold the shift key and use the Transform menu to select the
command Iterate to Depth. In the new window, select point A' for the image of point A.
8. Click + or – on parameter n to change the type of regular polygon.
Example 3
Using iterations, construct a regular polygon and all its diagonals from one vertex.
Solution:
1. Construct parameters n i t1. (Parameter n represents the number of vertices. We need the
parameter t1for calculations. The value of parameter t1 is 0.)
2. Draw points S and A.
3. Calculate n · n.
4. Calculate t1+ 1.
360° ⎞
5. Calculate ⎜⎛
⎟ ⋅ (t + t 1 ) .
⎝ n ⎠
360° ⎞
6. Rotate point A around point S by the angle ⎜⎛
⎟ ⋅ (t + t 1 ) . Name the new point A'.
⎝ n ⎠
7. Draw segment AA ' .
8. Select point A and parameter n. Hold the shift key and use the Transform menu to select the
command Iterate to Depth. In the new window, select point A' for the image of point A.
9. Select parameters t1 i n · n. Hold the shift key and use the Transform menu to select the
command Iterate to Depth. In the new window, select calculation t1 + 1 for the image of
the parameter t1.
10. Click + or – on parameter n to change the type of regular polygon.
Example 4
Using iterations, construct a regular polygon and all its diagonals.
Solution:
1. Draw points S and A.
2. Construct parameters n and t1. (The value for parameter t1 is 0.)
3. Calculate n · n.
4. Calculate t1 + 1.
(t + t 1 ) ⎞ ⎛ 360° ⎞
5. Calculate ⎛⎜
⎟.
⎟⋅⎜
⎝ n ⎠ ⎝ n ⎠
360° ⎞
6. Calculate ⎛⎜
⎟ ⋅ (t + t 1 ) .
⎝ n ⎠
7. Create the Action button Hide/Show to hide/show point A.
(Select point A and use the Edit menu to select the command Action Buttons and then the
command Hide/Show.)
(t + t 1 ) ⎞ ⎛ 360° ⎞
8. Rotate point A around point S by the angle trunc ⎛⎜
⎟ . Label the new point A'.
⎟⋅⎜
⎝ n ⎠ ⎝ n ⎠
9. Using the Action button Hide/Show, hide point A.
10. Insert the Action button Hide/Show to hide/show point A'.
11. Using the Action button Hide/Show, hide point A'.
12. Using the Action button Hide/Show, show point A.
360° ⎞
13. Rotate point A around point S by the angle ⎛⎜
⎟ ⋅ (t + t 1 ) . Label the new point A''.
⎝ n ⎠
14. Using the Action button Hide/Show, hide point A and show point A'.
15. Draw segment A ' A '' .
16. Using the Action button Hide/Show, hide point A' and show point A.
17. Select parameters t1 and n · n. Hold the shift key and use the Transform menu to select the
command Iterate to Depth. In the new window, select calculation t1+ 1 for the image of
parameter t1.
18. Click + or - on parameter n to change the type of regular polygon.
(101 Project Ideas)
Fig. 6 An example of student work on enrichment activity Iterations (By clicking + or – on
parameter one can change the type of a polygon.)
The activities used in our model create a motivating and engaging environment where
technology allows students to discover mathematics on their own and construct their own
understanding. The use of technology enables students to easily notice important properties of
objects, work at their own pace, focus on important concepts and build geometrical
knowledge. In addition, the use of technology enables teachers to actively involve students in
the learning process and individualize instruction. Therefore, it improves the teaching and
learning of geometry.
References
Serra, M. (2008). Discovering Geometry An Investigate Approach. Key Curriculum Press,
Emeryville, CA.
de Villiers, M. (2003). Rethinking Proof with The Geometer’s Sketchpad. Key Curriculum
Press, Emeryville, CA.
101 Project Ideas for The Geometer's Sketchpad. Key Curriculum Press. Available:
http://www.dynamicgeometry.com/General_Resources/101_Project_Ideas.html (19/09/2010)